[Grammar] "I would like to walk outside rather than to stay at home." is wrong?

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"I would like to walk outside rather than to stay at home." is wrong?

Hello,

I am a middle school student in Korea. Last week, I took a mid-exam and just wanted to check out one of English questions.

The question is to translate a Korean sentence into English and 5 points are allotted to this.
The answer that my English teachers defined is:
"I would like to take a walk rather than stay at home."
In the meantime, my answer was "I would like to walk outside rather than to stay at home."

Teachers say that my answer is completely incorrect for two reasons. First, 'walk outside' is not equal to 'take a walk', and second, 'to' is not supposed to be followed by rather than.

Is there anyone who can advise if my answer is completely wrong or a little bit worth even partial points?
I don't still understand why 'walk outside' cannot be used instead of 'take a walk'.

Re: "I would like to walk outside rather than to stay at home." is wrong?

Thank you for your comments. My understanding from your comments is it should be okay to use it like what I answered.

However, my English teacher (she is a Korean) finally decided to mark me down. no points at all, even a partial points.

Too sad and worried about how to learn English from onward. I feel like I'm getting afraid of using English as a second language here in Korea because they teach English like 'math'. They seem to consider English grammar something like math formular. Even though I make even a small piece of mistake in grammar and it has no problem in communication, they just judge it is completely wrong. I believe this is a serious problem of the way to teach English in Korea.